2. RECORD

4. Subspaces of Vector Spaces:
Check to see if there are any properties inherited from V:

5. Subspace Criterion in two forms:

6. More Examples:
The set of all symmetric matrices.
The set of all lower triangular matrices.
The set of all upper triangular matrices.
The set of all diagonal matrices.

7. Example: Subset of the Vector Space That is Not a Subspace

8. Example: Lines and Planes Through the Origin as Subspaces of R2 and R3

10. The Intersection of Subspaces

11. The Span of the Set of Vectors

12. Example: Spaces Spanned by One or Two Vectors in 3-space
Span of a subset in V is a subspace of V:

13. Exercises:
Determine if the following are subspaces of Mnxn.

15. Linear independence:
Examples:

16. Establishing dependence/independence
Exercises:

18. Revisit the example from page 3:

19. Results about dependent and independent sets:

20. Exercises:

22. Geometric Interpretation of Linear Dependence: Sets of Two Vectors

23. Geometric Interpretation of Linear Dependence: Sets of Three Vectors

26. Basis for the Vector Space: Definition

28. Exercises:

30. Exercises:

31. Dimension of the Vector Space:

32. Coordinate Representation of Vectors: